(PHH98 2d-Slide#15 \(8 of 8\)) - Atmospheric Dynamics Group

BALANCED FLOW:
A PROGNOSTIC EQUATION
(PHH lecture 2)
3-D rotating stratified flow
Consider 3-D fluid with density stratification.
Make hydrostatic approximation:
DOMINANT BALANCE IN VERTICAL MOMENTUM
EQUATION IS BETWEEN
VERTICAL PRESSURE GRADIENT
GRAVITATIONAL FORCE
(Can be argued from
vertical length scales

Thermal wind relation
Geostrophic balance: f × u = − 1 ρ0
∇p
Hydrostatic balance: − ∂p
∂z − ρg = 0
[Note: Boussinesq system with geometric height used
as a vertical co-ordinate unless otherwise stated]
Thermal wind balance:
f × ∂u
∂z = g
ρ0
∇ρ
f
u
heavy (cold)
light (warm)
N.B. Meteorology: maps of ‘thickness’ may be used to
infer the change in wind with height across a finite
layer. Satellite observations of temperature are
used to infer stratospheric winds.
Oceanography: observations of ρ are sometimes
used to infer velocities – requires ‘level of no motion’
Quasi-geostrophic flow
PHH 2 / 3
We seek a predictive equation for the time evolution of a
flow that is always close to geostrophic balance
Scale analysis of (hydrostatic) equations
∂u
∂t
+ u.∇u + f × u = − ∇p
ρ0
U
T
1
f T
U 2
L
U
f L
fU
1 relative sizes
‘ Rossby number’ Ro
1
f T or U f L
Small Rossby number requires
time scale > ≈ few days (at midlatitudes) and
U

• Geostrophic balance gives us no predictive equation
for the motion (at least if f is constant) because mass
continuity follows identically from the geostrophic
flow
• Need to consider the small departures from
geostrophic balance
Effects of spherical geometry:
• For large scale motion in the atmosphere and the
ocean we need to take account of some of the effects of
sphericity
• Rossby’s great insight was that the most important
effect is the latitudinal variation of the Coriolis
parameter:
y latitude co-ordinate
f = f0 + βy f0 , β constants PHH 2 / 5
[β-plane approximation]
f0 = 2Ωsin φ 0
β = 2Ω a cos φ 0
⎫
⎪
⎬
⎪
⎭
where φ0
is latitude
Derivation of quasi-geostrophic equations
[For careful mathematical derivation see Pedlosky
(Chapter 6)]
Consider primitive equations on a β-plane, and write
ρ = ρ ′ + ρb( z), p = p ′ + pb(z), where ρb is a ‘ basic state’
stratification and pb is the associated pressure field in
hydrostatic balance
∂
∂t u h + u.∇u h + (f 0 + βy)k × u h = − 1 ρ0
∇h p ′
∇.u = 0
⎛
⎜
⎝
u h = (u,v,0)
u = (u,v, w)
⎞
⎟
⎠
− ∂ p ′
∂z − ρ ′ g = 0
∂ ρ ′
∂t + (u.∇) ρ ′ + w ∂ρ b
∂z = 0
Divide velocity into geostrophic and ageostrophic parts
u = u g + u a
f0 k × u g = − 1
ρ0
∇ p ′
(defines geostrophic velocity)
w = wg + w a , but ∇.u = 0 ⇒ ∂w g
∂z = 0 ⇒ w g = 0
PHH 2 / 6

Ageostrophic part of equations
∂
∂t (u g + u a ) h + (u g + u a ).∇(u g + u a ) h
+ f0k × ua + βyk × (u g + u a) = 0
∇.u a = 0
∂ ρ ′
∂t + (u g + u a ).∇ ρ ′ + w a
dρb
dz = 0
− ∂ p ′
∂z − ′
This term retained – implicit
assumption that basic state vertical
density gradient is strong
From vorticity equation
ρ g = 0
using continuity equation for u a,
∂
∂t ζ g + u g .∇ζ g + βv g = − f 0
⎧
⎨
⎩
∂ua
∂x + ∂v a
∂y
⎫
⎬
⎭
= − f0
∂
∂z
⎧
⎪
⎨
⎩ ⎪
1
g ∂ρ b
∂z
⎡
⎢
⎣
∂
∂t
⎛
⎜
⎝
∂ p ′
∂z
⎞
⎟ + u g
⎠
.∇ ⎛
⎜
∂ p ′
⎝ ∂z
⎞
⎟
⎠
⎤
⎥
⎦
⎫
⎪
⎬
⎭ ⎪
then density equation
PHH 2 / 7
Define ψ = p ′/ f0 ρ0
(streamfunction for
quasi-geostrophic motion)
so ug = − ∂ψ
∂y , v g = ∂ψ
∂x , ρ ′ = − f 0ρ0
g
∂ψ
∂z
It follows that
⎧
⎨
⎩
∂
∂t + u g.∇
⎫
⎬
⎭ q g = 0
where
N 2 ( z) = − g
ρ0
dρb
dz
qg = ∂2 ψ
∂x 2 + ∂2 ψ
∂y 2 + ∂ ∂z
⎛
⎜
⎝
2
f0
N 2 ( z)
∂ψ
∂z
⎞
⎟
⎠
+ βy
quasi-geostrophic potential vorticity
Quasi-geostrophic p.v. is conserved following the
geostrophic motion (which is horizontal)
[Rossby–Ertel PV is conserved following the exact motion
– can be shown quite independently of any small Ro
assumption (see later).]
PHH 2 / 8

Boundary conditions for
quasi-geostrophic flow
Side boundaries: no normal flow
Top and bottom boundaries: vertical velocity zero (or set
by topography or ‘Ekman pumping’) – deduce from
density equation
⎧
⎨
⎩
∂
∂t + u g . ∇
⎫
⎬
⎭ ρ ′ = −w dρ b
dz
hence
known
⎧
⎨
⎩
∂
∂t + u g . ∇
⎫
⎬
⎭
∂ψ
∂z = − N 2 w
f0
on top or
bottom
boundaries
Top and bottom boundary conditions take the form of
prognostic equations for surface density or temperature
Similarity to two-dimensional flow:
In 2-D incompressible flow the component of vorticity
perpendicular to the plane of motion, q, is conserved
and, in terms of a streamfunction ψ, the vorticity
equation takes the form
PHH 2 / 9
⎧
⎨
⎩
∂
∂t + u.∇
⎫
⎬
⎭ q = 0
where q = ∂2 ψ
∂x 2 + ∂2 ψ
∂y 2 + βy
⎛
and u = (u, v) = ⎜ ⎝ − ∂ψ
∂y , ∂ψ
∂x
⎞
⎟ .
⎠
[Note that in non-GEFD applications, e.g. aerodynamics,
β would be taken to be zero]
A good deal of insight into the behaviour of solutions of
the q.-g. p.v. equation may therefore be gained from the
theory of 2-D vortex dynamics (MEM lectures and
various computer demonstrations).
PHH 2 / 10

Different contributions to q g
(and its horizontal gradients)
3-D flow:
qg = ∂2 ψ
∂x 2 + ∂2 ψ
∂y 2 + ∂ ∂z
⎛
⎜
⎝
f0 2
N 2 ( z)
∂ψ
∂z
⎞
⎟
⎠
+ βy
1 2 3
1 Relative vorticity
2 Vortex stretching term (associated with
vertical density gradient)
3 Planetary vorticity
1 and 2 are comparable if f0 L ≈ ND
(Prandtl’s ratio of scales)
D
L
( f0 / N < ≈
10−2 for both
atmosphere and ocean,
i.e. thermocline)
1 and 3 are comparable if U ≈ βL2
βL 2 ≈ 10 −1 ms −1 if L ≈ 100 km
10 ms −1 if L ≈ 1000 km
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Numerical simulations of geostrophic turbulence
(McWilliams, JFM 1989)
t = 10
t = 20
PHH 2 / 12

Numerical simulations of geostrophic turbulence
(McWilliams, JFM 1989)
Isovorticity surfaces cyclones light
anticyclones dark
f/N isotropy with small deviations (e.g. due to fact that
small vertical scales have little relative vorticity)
PHH 2 / 13
Understanding evolution of QG turbulence is an ongoing
research problem
(McWilliams et al 1994) find 'columns'as long-time state
A is initial condition, time increases through B,C,D
but see Dritchel & Ambaum (1997) PHH 2 / 14

Summary
The quasi-geostrophic equations are a convenient set
to describe motion at small Rossby number which
remains close to geostrophic balance. [But they are
not quantitatively accurate for modelling realistic
atmosphere or oceanic flow.]
Evolution equation Invertibility principle
qg (quasi - geostrophic
p.v.)
qg and ∂ψ
∂z
(boundaries)
(non-local
operator)
∂ψ
∂z
(temperature at
boundaries)
flow, temperature etc.
in geostrophic balance
GENERALIZATION (see Hoskins et al. 1985)
P (Rossby–Ertel p.v.)
P and θ (boundary)
or σθ (boundary)
θ or σθ (potential temperature
or potential density
at boundaries)
flow, temperature etc.
in balance
(non-local
operator)
PHH 2 / 15